Perron Vector Research Articles

Some properties concerning Perron vectors of weakly irreducible nonnegative tensors, and their application to rigorous enclosure

We clarify some properties concerning Perron vectors of weakly irreducible nonnegative (WIN) tensors, and multi-linear systems with non-homogeneous left-hand side whose solutions are sub-vectors of the Perron vectors. Based on the properties, we propose an algorithm for computing interval vectors containing the Perron vectors. This algorithm is applicable to all the WIN tensors, whereas the algorithms previously developed by the author are applicable only to a subclass of the WIN tensors. Numerical results show efficiency of the proposed algorithm.

Large time behavior of nonautonomous linear differential equations with Kirchhoff coefficients

Nonautonomous linear ordinary differential equations with Kirchhoff coefficients are considered. Under appropriate assumptions on the topology of the directed graphs of the coefficients, it is shown that if the Perron vectors of the coefficients are slowly varying at infinity, then every solution is asymptotic to a constant multiple of the Perron vectors at infinity. Our results improve and generalize some recent convergence theorems.

On the Convergence Properties of a Distributed Projected Subgradient Algorithm

A weight-balanced network plays an important role in the exact convergence of distributed optimization algorithms, but is not always satisfied in practice. Different from most of existing works focusing on designing distributed algorithms, we analyze the convergence of a well-known distributed projected subgradient algorithm over time-varying general graph sequences, i.e., the weight matrices of the network are only required to be row stochastic instead of doubly stochastic. We first show that there may exist a graph sequence such that the algorithm is not convergent when the network switches freely within finitely many graphs. Then to guarantee its convergence under any uniformly jointly strongly connected graph sequence, we provide a necessary and sufficient condition on the cost functions, i.e., the intersection of optimal solution sets to all local optimization problems is not empty. Furthermore, we surprisingly find that the algorithm is convergent for any periodically switching graph sequence, but the converged solution minimizes a weighted sum of local cost functions, where the weights depend on the Perron vectors of some product matrices of the underlying switching graphs. Finally, we consider a slightly broader class of quasi-periodically switching graph sequences, and show that the algorithm is convergent for any quasi-periodic graph sequence if and only if the network switches between only two graphs. This work helps us understand impacts of communication networks on the convergence of distributed algorithms, and complements existing results from a different viewpoint.

Eigenvalues for stochastic matrices with a prescribed stationary distribution

Given a vector $0<w \in \mathbb{R}^n$ whose entries sum to $1$, the region $\sigma_\mathcal{S}(w)$ in the complex plane consisting of all eigenvalues of all stochastic matrices having $w^\top$ as a left Perron vector is considered. Some general observations about this region are made, it is proven that $\bigcap_{w \in \mathbb{R}^n, w>0, w^\top \mathbf{1} =1} \sigma_\mathcal{S}(w) =[0,1],$ and a characterization is given of the vectors $w$ such that $\sigma_\mathcal{S}(w)$ contains an element $\lambda \ne 1$ with $|\lambda|=1.$ The corresponding problem for reversible stochastic matrices with given left Perron vector is also considered, as is the corresponding region $\sigma_\mathcal{R}(w),$ which is a subset of $[-1,1].$ Under a mild hypothesis on $w, $ it is proven that the smallest element of $\sigma_\mathcal{R}(w)$ corresponds to a reversible stochastic matrix whose graph is a tree with a loop at one vertex. A general lower bound on the eigenvalues of reversible stochastic matrices with given left Perron vector is also given, as is a complete description of $\sigma_\mathcal{R}(w)$ when $w$ has two or three entries.

Two-step Noda iteration for irreducible nonnegative matrices

ABSTRACT In this paper, we present a two-step Noda iteration for computing the Perron root and Perron vector of an irreducible nonnegative matrix by successively solving two linear systems with the same coefficient matrix. For every positive initial vector, the two-step Noda iteration always converges and has a cubic asymptotic convergence rate. As an application, the two-step Noda iteration is applied to compute the smallest eigenpair of irreducible nonsingular M-matrices. Numerical examples are provided for testing the proposed method.

Maximum principal ratio of the signless Laplacian of graphs

Let G be a connected graph and Q(G) be the signless Laplacian of G. The principal ratio γ(G) of Q(G) is the ratio of the maximum and minimum entries of the Perron vector of Q(G). In this paper, we consider the maximum principal ratio γ(G) among all connected graphs of order n, and show that for sufficiently large n the extremal graph is a kite graph obtained by identifying an end vertex of a path to any vertex of a complete graph.

Entrywise lower and upper bounds for the Perron vector

Let an irreducible nonnegative matrix A and a positive vector x be given. Assume αx≤Ax≤βx for some 0<α≤β∈R. Then, by Perron-Frobenius theory, α and β are lower and upper bounds for the Perron root of A. As for the Perron vector x⁎, only bounds for the ratio γ:=maxi,j⁡xi⁎/xj⁎ are known, but no error bounds against some given vector x. In this note we close this gap. For a given positive vector x and provided that α and β as above are not too far apart, we prove entrywise lower and upper bounds of the relative error of x to the Perron vector of A.

Geometric bounds for convergence rates of averaging algorithms

We develop a generic method for bounding the convergence rate of an averaging algorithm running in a multiagent system with a time-varying network, where the associated stochastic matrices have a time-independent Perron vector. The resulting bounds depend on geometric parameters of the dynamic communication graph such as the weighted diameter or the bottleneck measure.As corollaries, we show that the convergence rate of the Metropolis algorithm in a system of n agents is less than 1−1/4n2 if the communication graph is permanently connected and bidirectional. We prove a similar upper bound for the EqualNeighbor algorithm under the additional assumptions that the number of neighbors of each agent is constant and the communication graph is not too irregular. In general, our bounds offer improved convergence rates for several averaging algorithms and specific families of communication graphs.

The dimension of eigenvariety of nonnegative tensors associated with spectral radius

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar, called Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. We prove that the dimension of the above projective eigenvariety is zero, i.e. there are finitely many eigenvectors associated with the spectral radius up to a scalar. We characterize a nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs.

Fast verified computation for positive solutions to [formula omitted]-tensor multi-linear systems and Perron vectors of a kind of weakly irreducible nonnegative tensors

Two fast numerical algorithms are proposed for computing interval vectors containing positive solutions to M-tensor multi-linear systems. The first algorithm involves only two tensor–vector multiplications. The second algorithm is iterative one, and generally gives interval vectors narrower than those by the first algorithm. We also develop two verification algorithms for Perron vectors of a kind of weakly irreducible nonnegative tensors, which we call slightly positive tensors. The first and second algorithms have properties similar to those of the two algorithms for the solutions to the M-tensor systems. We clarify relations between slightly positive tensors and other tensor classes. Numerical results show efficiency of the algorithms.

Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Some improvements on the Ky Fan theorem for tensors

The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

Convergence in nonautonomous linear differential equations with Kirchhoff coefficients

Abstract A class of nonautonomous linear ordinary differential equations is considered. The coefficient matrices are Metzler matrices with zero column sums such that their directed graphs have a common directed spanning tree. It is shown that if the off-diagonal elements of the coefficients are uniformly positive along the common directed spanning tree, then under mild additional assumptions the convergence of the Perron vectors of the coefficient matrices implies that all solutions tend to a finite limit at infinity. The value of the limit can be expressed in terms of the initial data.

Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

Functions and eigenvectors of partially known matrices with applications to network analysis

Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for the network. The present article addresses the problem of evaluating matrix functions, as well as computing an approximation to the left Perron vector, when only some of the columns and/or some of the rows of the adjacency matrix are known. Applications to network analysis are considered, when only some sampled columns and/or rows of the adjacency matrix that defines the network are available. A sampling scheme that takes the connectivity of the network into account is described. Computed examples illustrate the performance of the methods discussed.

Some spectral inequalities for connected bipartite graphs with maximum [formula omitted]-index

Let G be a graph, whose adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017 Nikiforov (2017) merged the A- and Q-spectral theories to proposed the Aα matrix: Aα(G)=αD(G)+(1−α)A(G), α∈[0,1]. Its largest eigenvalue is called the Aα-index of G. In this paper, we focus on specifying the property of the Aα matrix. On the one hand, we show that among the set of connected bipartite graphs of fixed order and size, the extremal graphs with the maximum Aα-index are the double nested graphs, i.e., bipartite chain graphs. On the other hand, we establish a series of inequalities respecting the entries of the Perron vector of Aα(G) of double nested graphs. Then we obtain some lower and upper bounds on the Aα-index of double nested graphs using the eigenvector and matrix techniques. Finally, we provide some computational results to compare these bounds.

Type the title of your paper here mathematical properties of n×n nonnegative matrix: case of irreducible Leslie matrix

Perron-Frobenius theorem describe five properties of irreducible nonnegative matrix. Leslie matrix is one of nonnegative matrix. Leslie matrix that used in this research is limited to the irreducible Leslie matrix. In previous research has been proven that irreducible Leslie matrix satisfies three properties in Perron-Frobenius theorem by spectral radius. This research completed the previous research, proving that irreducible Leslie matrix has a unique Perron vector and satisfies Collatz Wielandt formula. Leslie matrix is a primitive matrix. It is used to calculate the number of populations in the future. Growth of population is interpreted by value spectral radius of Leslie matrix.

Bearing Condition Monitoring based on the Indicator Generated in Time-frequency Domain

Most condition monitoring systems rely on system-driven generation of indicators or features for early fault detection. However, this strategy requires the prior knowledge on the system kinematics and/or exact structure parameters of monitored system. To address this problem, this paper presents a novel condition monitoring framework where the condition indicator is generated via data-driven method. In this framework, the time-frequency periodogram is extracted from raw vibration signal first. Then, the acquired time-frequency periodogram is mapped by pseudo Perron vector, which is learned from vibration data, to generate the condition indicator. Finally, the bearing can be monitored via analyzing this indicator using gaussian based control chart. Based on experimental results on a publicly-available database, we show the effectiveness of presented framework for early fault detectionin the continuous operation of rolling bearing, indicating its great potentials in real engineering applications.

Perron vector analysis for irreducible nonnegative tensors and its applications

<p style='text-indent:20px;'>In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.

Ergodicity in nonautonomous linear ordinary differential equations

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.